Sane, making his first league start for Pep Guardiola's side this term, opened the scoring after one minute and 36 seconds with a simple tap-in from Fernandinho's pass.
The brilliant David Silva doubled the hosts' lead before the break, with Raheem Sterling all but securing the win two minutes into the second half.
A fourth win in five games keeps City within touching distance of league leaders Liverpool and Chelsea, who have taken maximum points so far.
City were rarely troubled by the newly promoted Cottagers, restricting the visitors to handful of half chances at Etihad Stadium.
The hosts, perhaps with one eye on Wednesday's Champions League opener against Lyon, eased off the pace in the second half but could have added to their lead with a number of opportunities falling to substitute Gabriel Jesus.
Sane enjoyed a glittering season as Manchester City won the title last term, scoring 10 goals and proving one of Guardiola's main attacking outlets, but he has been restricted to a spot on the bench so far this campaign.
It took the winger, who also surprisingly missed out on a place in Germany's World Cup squad in the summer, less than two minutes to find the net on his return.
The hosts punished Fulham's Jean Michael Seri for giving the ball away in midfield as Fernandinho broke forward and teed up 22-year-old Sane inside the six-yard-box.
It was the quickest Premier League goal of the season so far and the culmination of a good few days for Sane, who said after the game he became a father this week.
City's only dropped points this season came at newly promoted Wolves but, having not lost at home to a promoted team in 11 years, they were always in complete control against Fulham and doubled the lead on 20 minutes.
Bernardo Silva's good work down the left was eventually finished off by David Silva's half volley that struck the bar on its way in - it was the Spaniard's 50th Premier League goal, becoming the fifth City player to reach that landmark after Sergio Aguero, Yaya Toure, Carlos Tevez and Edin Dzeko.
Sterling had also hit the bar in the first half as he and Sane caused problems out wide, but the England forward got his reward for a strong performance within two minutes of the restart as he tapped in from Aguero's low cross.
Fulham boss Slavisa Jokanovic said before the match that while his side wanted to use "similar weapons" to City, they must follow their own style of football.
The visitors brought two attacking full-backs into their line-up with Cyrus Christie on the right and teenager Ryan Sessegnon, often utilised as a winger, making his first start as a left-back since before Christmas.
But that left Fulham's central defenders exposed, with City exploiting the space left behind by 18-year-old Sessegnon as Aguero set up Sterling with a low cross from the right for City's third.
Fulham, promoted via the play-offs last season, looked threatening in the rare spells they were able to bring wide forwards Andre Schurrle and Luciano Vietto into the game, with World Cup winner Schurrle forcing a smart save from Ederson in the first half.
However, prolific forward Aleksandar Mitrovic, who has scored 24 goals for club and country in 2018, remained isolated and was restricted to only one shot on target from long range late in the second half.
It was another lesson for Fulham following their return to top-flight football, with the Cottagers having now conceded 12 goals in five games - the worst defensive record in the division.
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